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Miller Indices and some useful relations

Miller indices of Plane

Procedure to determine miller indices of plane: (1) Choose origin in such a

way which lies outside the plane of interest (choice of origin is arbitrary) (2)

Find the intercepts of the plane on the three co – ordinate axes (3) Take

reciprocals (4) Convert into smallest integers in the same ratio (5) Enclose

them in parentheses without camos eg : (h k l)

( ) denotes plane, { } denotes family of planes

Useful aspects about miller indices for planes (1) A plane and its negative

are identical eg: (0 1 0) = (0 -1 0) (2) Planes and their multiples are not

identical as planar densities and packing factors are different (3) Family of

planes will have same type of atomic packing but not all members of family

are parallel to one another

Miller indices of direction Directions in crystal are specified in a shorthand

vector notation. Let a vector r represents a direction in a crystal. The miller

indices are simply the vector components of direction resolved along each

of the co – ordinate axes and reduce to smallest integers i.e the components

of the vector along 3 axes are determined as multiples of the unit vector

corresponding to each direction.

[ ] denotes direction, < > denotes family of directions

Useful aspects about miller indices for directions (1) A direction and itsnegative are not identical, [1 0 0] ≠ [-1 0 0] same line but opposite direction

(2) A direction and its multiple are identical, [1 1 0] = [2 2 0], but should

not be reduced to lowest integers (3) Crystal directions of family are not

necessary parallel to one another (4) Crystal plane and a crystal direction

normal to it have same indices i.e [1 1 1] (1 1 1)

Miller – bravias indices For hexagonal crystals a four digit notation h k i l

known as Miller – bravias indices is used. The use of such a notation

enables crystallographically equivalent planes or directions in hexagonal

crystals to be denoted by the same set of indices. 3 axes a1, a2, a3 are

coplanar and lie on the basal plane of the hexagonal prism with a 120° angle

between them. The fourth axis is the c axis perpendicular to the basal plane.

The indices of plane or direction are calculated similar to that in miller

indices. For 3 coplanar vectors h+k = -i

Inter planar spacing The distance or spacing between the plane and a

parallel plane passing through the origin. In case of cubic system it is given

by

Angle between planes or directions

For cubic crystals, the angle between two planes (h1 k 1 l1) and (h2 k 2 l2) or

two directions [h1 k 1 l1] and [h2 k 2 l2] is

Line of intersection [h k l] of two planes (h 1 k 1 l1) and (h2 k 2 l2) is cross

product of two planes.

h = k 1l2 – l1k 2 k = l1h2 – h1l2 l = h1k 2 – k 1h2

The direction [h1 k 1 l1] lies in the plane (h2 k 2 l2) if h1h2 + k 1k 2 + l1l2 =

0.

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Labels: Basics

Interstitial Voids The empty space that is unoccupied by the atoms in a

closed packed system.

Types: (1) tetrahedral (2) octahedral

Tetrahedral void

This forms when an atom is put in the valley formed by three spheres of a

closed – packed plane

The named has been derived because regular tetrahedron is formed when

he centers of 4 atoms are joined

Octahedral void It is surrounded by 6 solvent atoms situated at 6 corners of a regular

octahedron

The name has been derived because of 8 equal faces of equilateral triangles

4 atoms in a plane (square based) and one on top and other at bottom

Iron-Iron carbide Phase Diagram

Pure iron has two crystalline forms, one BCC, commonly called α - iron

which remains stable from low temperatures upto 910°C (1414°F) when it

changes to FCC called γ – iron. The γ - iron remains stable upto 1394°C

(2554°F), when it reverts to BCC form now called as δ - iron, which is

stable upto the melting point of iron (1539°C or 2802°F).All the allotropicchanges give off heat (exothermic) when iron is cooled and absorb heat

(endothermic) when iron is heated.

Effect of pressure on allotropy of iron

Increse in pressure lowers the α - Fe to γ - Fe transition temperature and

increses the γ - Fe to δ - Fe traqnsition temperature. This is according to the

lechatlier’s principle as volume of FCC (γ - Fe) is lower than that of BCC.

A volume change of FCC to BCC is 8.8%.

Iron – Iron carbide diagram

The temperature at which allotrophic change (critical temperatures) takes place is influenced by alloying elements. The curie temperature is not

effected by alloying elements. Carbon is the most common alloying

element in the iron which significantly affects the allotrophy, structure and

properties of iron. Conventionally, the complete Fe – C diagram should

extend from 100% Fe to 100 % carbon (graphite), but it is normally studied

upto 6.67% carbon (Fe3C) because iron alloys of practical industrial

importance contain not more than 5% carbon. Fe – Fe3C is not a true

equilibrium diagram, since Fe3C (meta stable) decomposes into Iron and

Carbon which take a very long time at room temperature and even at 700°C

it takes several years to form graphite. When the carbon content becomes

more than solubility limits of iron though carbon should be present as

Graphite (lower free energy than cementite) , yet cementite forms beacause

the formation of cementite is most probable kinetically i.e. it is easier to

from it, as only 6.67% C has to diffuse to segregate to form cementite

whereas 100% C segregation is required to nucleate graphite. .

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Definition of structures or phases in Fe – Fe3C diagram

Ferrite: it is an interstitial solid solution of carbon in α - iron (BCC). The

maximum solubility of carbon in ferrite is 0.02 wt% at 727°C and the

minimum is 0.00005 wt% at 20°C. the size of the largest atom that can fit in

octahedral void is 0.19 A°, which is much smaller than carbon atom

(0.71°A°). so the solubility is exremely limited. It is soft and ductile. Ferrite

is ferromagnetic upto 768°C becomes paramagnetic above this temperature.

Austenite: it is an interstitial solid solution of carbon in γ - iron (FCC). The

maximum solubility of carbon is 2.1 wt% at 1146°C which decreses to 0.77wt% at 727°C. the size of the largest atom that can fit in octahedral void is

0.52 A°. correspondingly the solubility is larger here compared to ferrite. It

is soft, ductile, malleable, tough and non-magenetic. It is stable above

727°C in plain carbon steels but can be obtained even at room temperature

by adding elements like Ni or Mn in steels.

δ - ferrite: it is an interstitial solid solution of carbon in δ - iron (BCC). The

maximum solubility of carbon is 0.09 wt% at 1495°C. it is paramagnetic. It

is high temperature version of α -iron.

Cementite (Fe3C): It is an interstitial intermetllic compound havinbg fixed

carbon content of 6.67wt%. it has a complex orthorhombic structure, with

12 Fe atoms and 4 C atoms per unitcell. High hardness, brittle, very low

tensile strenght and high compressive strength. It is the hardest phase thatappears on the phase diagram.

Ledeburite: it is eutectic mixture of austenite and cementite. It contains

4.3wt% C and is formed at 1146°C. this is very fine mixture.

Pearlite: it is eutectoid mixture of ferrite and cementite containing 0.8wt%

C and is formed at 727°C. it is a very fine platelike or lamellar mixture.

Invariant reactions in Fe – Fe3C diagram

1. Peritectic reaction

Composition wt% 0.09 0.53 0.17

2. Eutectic reaction


3. Eutectoid reaction


Critical temperature in Fe – Fe3C diagram

The temperatures at which phase transformations occurs during heating or

cooling an alloy. certain symbols are used to denote the critical temperature

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Labels: Basics

in steels. The upper and lower critical lines under equilibrium are indicated

by Ae3 and Ae1 etc.

It is found that in actual practice the critical line on heating and the critical

line on cooling are not occur at same temperature. The critical line on

heating is always higher than the critical line on cooling. The former is

denoted by Ac and the later is denoted by Ar. A for arret (means arrest), C

for chauffage (means heating), R for Refroidissement (means cooling), e for

equilibrium.

If extremely slow rates of heating or cooling are employed then critical

temperatures are nearly equal i.e. Ac1 = Ar1 = Ae1.

The curie temperature (magnetic to non-megnetic change) of cementite is

called A0. Ae1 or A1 is eutectoid tempersture line (727°C). Ae2 or A2 is

curie temperature line (768°C) and this is constant for all Fe – C alloys.

Hypo-eutectoid side

Upper critical temperature (Ae3 , Ac3 , Ar 3): It is the temperature at which

Austenite to ferrite transformation begins on cooling (or) at which ferrite to

austenite transformation ends on heating. This is denoted by A 3 line. (Ac3

> Ar 3)

Lower critical temperature: It is the temperature at which Austenite to

ferrite transformation ends on cooling (or) at which ferrite to austenite

transformation starts on heating. (Ac1 > Ar 1).

Hypo-eutectoid sideLower critical temperature (Ae3,1 Ac3,1 Ar 3,1) : It is the temperature at

which precipitation of cementite from austenite ends uopn cooling (or) at

which dissolution of cementite in austenite begins upon heating.

Upper critical temperature (Aem , Acm , Ar m): It is the temperature at which

precipitation of cementite from austenite begins uopn cooling (or) at which

dissolution of cementite in austenite ends upon heating.

Effect of alloying elements on the Fe – C diagram

Ferrite stabilizers: some alloying elements tend to stabilize the ferrite

phase in preference to austenite. Many of these elements have same crystal

structure as ferrite (BCC). They reduce the extent of the austenite area on

the equilibrium diagram by forming a gamma loop. Austenite is enclosedwithin the loop. eg: Cr, Si, Mo, W, V, Ti etc.

Austenite stabilizers: These enlarge the area of the austenite phase on the

phase diagram. critical amount of these alloying elements results in

Austenite even at room temperture. eg: Mn, Ni, C, N etc.

Effect on eutectoid temperature and composition

Ferrite stabilizers raises the eutectoid temperatute to above 727°C,

Austenite stabilizers lowers the euctectoid tempersture to below 727°C.

Both Ferrite and Austenite stbilizers decrease the eutectoid composition

from 0.77% to lower values.

Selective leaching

Selective leaching is removal of one element from solid alloy by corrosion

process. Parting is metallurgical term sometimes applied but selective

leaching is preferred. The term dealloying is frequently used and is

preferred by some corrosionists. Dimensional changes do not occur. The

most common example is removal of Zinc in brass . Similar process occurs

in other alloy systems in which Al, Fe, Co, Cr and other metals are

removed.

Advantages of selective leaching

(1) Enrichment of silicon observed in the oxide film on stainless steels

results in better passivity and resistance to pitting (2) Preparation of Raney

nickel catalyst by selectively removing aluminium from Al-Ni alloy by

action of caustic.

Dezincification

Two general types (i) uniform or layer-type (seems to occur in high brasses

i.e. high Zn content) (2) localized or plung-type (seems to occur in low

brasses i.e. low Zn content). The dezincified portion is weak, permeable,

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Labels: Basics

porous, brittle and possesses little aggregate strength. Hence addition of

zinc to copper lowers the corrosion resistance of copper. Two mechanisms

have been proposed (1) Zinc is dissolved, leaving vacant sites in the brass

lattice structure (not proven) (2) Brass dissolves, zinc ions stay in the

solution and copper plates back on (commonly accepted).

Prevention: (1) reducing aggressiveness of medium (i.e. oxygen removal)

or by cathodic protection. but these not economical. (2) Addition of Sn to

Admirality brass (70Cu-30Zn). (3) add inhibitors like As, Sb, P.

Graphitization Gray cast irons some times shows selective leaching in

relatively mild environments. the name is given based on the surface layer has appearance of graphite. it is misnomer to call this as graphitization as

graphite presents in the material before corrosion. This is also called as

graphitic corrosion. selective leaching of Iron or steel matrix leaving the

graphite network in gray cast iron (presence of graphite flakes). Graphite is

cathodic to ferrite, (as amount of carbon is less in ferrite) and a galvanic cell

exists, which results in dissolution of iron leaving a porous mass consisting

graphite, voids and rust and loses its strength. Surface shows rusting This

corrosion does not occur in nodular or malleable (absence of graphite

flakes) and white cast iron (no free carbon).

Cathodic and Anodic protection

Cathodic protection This is method of reducing or preventing corrosion of

a metal by making it a cathode in the electrolytic cell. This can be achieved

by means of an externally impressed current or sacrificial anode. An

electrolyte is needed to ensure the passage of current through the part to be

protected. This is effective only in soils or aqueous media where part to be

protected is immersed. It is not effective in the atmosphere.

(1) Impressed –current method an external DC power supply is connected

to the metal be protected. The negative terminal of power supply is

connected to the part to be protected and the positive to an Auxiliary or

inert anode eg: graphite. Steel scrap, Al, Si-Fe are also can be used. Si-

Fe and graphite are suitable for ground-beds-buried or sea-bed for

marine projects.

Applications: pipe-lines, underground cables of Al, Pb; storage tanks,

heat-exchangers, steel-gates exposed to sea water, hulls of ships,highways and bridges.

(2) Sacrificial anode (or galvanic coupling) in this metal which has more

negative electrode potential than the structure to be protected is

connected electrically to the part or structure to be protected. The

structure is protected at the sacrifice of another metal. Mg alloys, Zn,

Al-5%Zn are widely used. These anodes are replaced as soon as

consumed.

Applications: under-water parts of ships, ship hull, underground pipes,

steel water tanks, water heaters, condenser tubes, oil-cargo-ballest tanks.

Galvanized sheet is sacrificial protection of steel (Zn on steel).

Anodic protection This is based on the formation of a protective film on

metals by externally applied anodic currents. An external current icrit

is

initially applied impressed on the metal so as to passivate it. Then the

current density is reduced to i passive and maintained at that value to ensure

the passive film does not dissolve. Material must exhibit passivity in

corrodent eg: Ni, Fe, Cr, Ti and their alloys. A potentiostat is used to

maintain the metal at a constant potential w.r.t a reference electrode. If the

control is lost temporarily and the potential strays into the anodic region, the

corrosion can be disastrously high. The primary advantage is its

applicability in extreme corrosive environments with low current

requirements.

Comparison of Anodic and Cathodic protection

Anodic Cathodic

Applicability Active-passivematals/alloys

All metals/alloys

Nature of corrosivemedium

Weak to aggressive Weak to medium

Cost: Installation Maintenance

HighVery low

LowMedium to high

Operating conditions Can be accuratelydetermine

Determined byempirical testing

Significance of Direct measure of Complex to indicate

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Labels: Basics

applied current protected corrosionrate

corrosion rate

Labels: Basics

Strenghtening from fine particles

Small second phase particles (Hard) distributed in a ductile matrix are a common

source of alloy strengthening.

Precipitation hardening is produced by solution treating (heating to a single phase

region) and quenching an alloy in which a second phase is in solid solution at the

elevated temperature but precipitates upon quenching and aging at room

temperature (natural aging) or at slightly higher temperature (artificial aging, 100-

200degrees). for this to occur the second phase must be soluble at an elevated

temperature but must exhibit decreasing solubility with decreasing temperature.

coherency between the precipitates and matrix is essential. The hardness increases

with the formation of GP zones (Guinier-preston) and the intermediate transition

precipitates.. A peak in hardness results due to critical dispersion of GP zones ,

further aging leads to decrease in hardness due to coarsening of precipitates

(incoherent). This is called over-aging. eg:Al-Cu (aerospace industry) and Cu-Be

(sparking tools in coal-mines).

The peak hardness depends on

1. Average particle size (fine particles)

2. Number of Particles (more finer particles)3. Inter particle distance (less)

Methods of studying precipitation

1.Mechanical properties During aging as the amount of precipitate increases with

time which increases the strength or hardness of the alloy. tension test or hardness

measurement can be used to know the changes in mechanical properties.

2. Electrical resistivity During aging the excess solute comes out gradually hence

strains in the crystal lattice decreases hence resistivity decreases .

3. X-ray diffraction Its application is to measure strain in the crystal. strains in the

lattice will decrease with time.

4. Electron Microscopy as precipitates are very small in size (few nano meters) we

have to use electron microscopy to observe the precipitation.

The fraction of second phase is limited by solubility limit. Higher supersaturation

causes faster precipitation. The degree of super saturation decreases with the

increase of aging temperature resulting in lower peak hardness at high temperatures.

since the amount of second phase is less and inter particle distance is high.

Reasons for hardening in precipitation-hardening

1. Internal strain-hardening by elastic coherency strains around GP zones.

2. Chemical-hardening due to precipitates being sheared (cut) by moving

dislocations.

3. Dispersion-hardening due to formation of loops of dislocations around

precipitates.

The requirement of a decreasing solubility with temperature places a limitation on

the number of useful precipitation-hardening alloy systems. Precipitation hardened

alloys can't be used at higher temperatures precipitates dissolve in the matrix at

higher temperature. To overcome these difficulties a dispersion strengthening is

developed.

Dispersion hardening The hard and strong foreign particles are dispersed in a

metal/alloy (matrix). Powder metallurgy is the best route to consolidate these

dispersion alloys. These particles are oxides, carbides, nitrides etc. Such alloys are

called dispersion strengthened alloys. The second phase alloys has very little

solubility in the matrix even at elevated temperatures. No coherency between the

second-phase particles and the matrix. These alloys are much more resistant to

recrystallization and grain growth than single-phase alloys. Second phase particles

donot dissolve even at high temperature.

Advantages

1. can be used for High temperature applications

2. we can use this for any alloy system

3. No limitation on the fraction or amount of dispersoid

Cold working and Annealing

Cold working is deformation carried out under conditions where recovery

processes are not effective. Hot working is deformation under conditions of

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temperature and strain rate such that recovery processes take place

simultaneously with the deformation.

Structural changes during cold working of polycrystalline metals and

alloys

(1) Changes in shape and size of grains: The equiaxed grains on

deformation are elongated in the direction of acting force i.e. stretched

in the direction of main tensile deformation stress–say, in the direction

of rolling or wire drawing.

(2) Changes in orientation of grains: Preferred orientation or texture of is

the state of severely cold worked metal in which certain crystallographic

planes of the grains orient themselves in a preferred manner with respect

to the direction of the stress (or maximum strain).

(3) Changes in internal structure of grains: during cold working around

15% of the work of the deformation gets absorbed in the material (rest is

lost as heat). This stored energy is the form of energy of crystal defects.

Plastic deformation increases the concentration of point defects. With

increase of cold working, the number of stacking-faults increases, thus

density of extended dislocations increases. The number of kinks, jogs,

dipoles, prismatic loops increase. The most important internal change of

structure is increase in density of dislocation from 106 – 108 cm-2 in

annealed state to 1010 – 1012 by moderate cold working.Effect of cold work on properties

Cold working or strain hardening is the increase in the stress required to

cause further slip because of previous plastic deformation. This is an

important industrial process that is used to harden metals or alloys that do

not respond to heat treatment. It changes various mechanical, physical and

chemical properties of metals and alloys.

With increase in amount of cold work, Ultimate Tensile Strength, Yield

Strength, Hardness increases but ductily (elongation and reduction in area)

decreases. Cold worked texture and mechanical fibering leads to Anisotropy

in in properties of materials. The ductility and impact toughness is much

lower in transverse section rather than in longitudinal section. As the

internal energy of cold worked state is high, the chemical reactivity of the

material increases i.e. the corrosion resistance decreases, and may cause

stress corrosion cracking in certain alloys. The rate of strain hardening

(slope of flow curve) is generally lower in HCP metals than cubic metals.

High temperatures of deformation also lower the rate of strain-hardening.

Annealing of Cold worked materials

In certain applications materials are used in the cold-worked state to derive

benefits of increased hardness and strength. The cold worked dislocation

cell structure is mechanically stable, but not thermodynamically stable. It is

necessary to restore the ductility to allow further cold deformation or to

restore the optimum physical properties such as electrical conductivity

essential for applications. The treatment to restore the ductility or electricalconductivity with a simultaneous decrease in hardness and strength is

Annealing (or Recrystallization annealing). It is heating cold worked metal

to a temperature above recrystallization temperature, holding there for some

time and then slow cooling.

The process of Annealing can be divided into three fairly distinct stages (1)

Recovery (2) Recrystallization (3) Grain growth. There is no change in

composition or crystal structure during annealing. The driving force for

recovery and recrystallization is the stored cold-worked energy, whereas for

grain growth is the energy stored in grain boundaries.

Recovery It is restoration of the physical properties of the cold worked

metal without of any observable change in microstructure. It is the

Annihilation and rearrangement of point imperfections and dislocations

without the migration of high angle grain boundaries. Recovery is initially

very rapid, and more when the annealing temperature is high. Electrical

conductivity increases rapidly toward the annealed value and lattice strain

measured using XRD is appreciably reduced. Properties those are sensitive


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to point defects are affected, and strength properties are not affected. With

increasing time at constant temperature the recovery becomes slower. The

greater the initial cold work, the more rapid is the initial rate of recovery.

The rate of recovery of fine grains is higher than that of coarse grains.

Polygonization one of the recovery processes which leads to rearrangement

of the dislocations, with a resultant lowering of the lattice strain energy. It is

a process of arranging excess edge dislocations in the form of tilt

boundaries, and the excess screw dislocations in the form of twist

boundaries, with the resultant lowering of the elastic strain energy. Climb

and slip of dislocations are essential for polygonization. The presence of

solute atoms in a metal reduces the rate of polygonization.

Recrystallization It is nucleation and growth of new strain-free crystals

from the cold worked metal. Kinetics of recrystallization resembles a phase

transformation. Two distinct nucleation mechanisms have been identified.

(1) Strain-induced boundary migration, where a strain-free nucleus is

formed when one of the existing grain boundaries into its neighbour, leaving

a strain-free recrystallized region. (2) new grains are formed in the regions

of sharp lattice curvature through subgrain growth. This seems to

predominate at high strains, with nuclei appearing at grain boundaries or at

inclusions or second phase particles. Mechanical properties change

drastically over a very small temperature range to become typical of the

annealed material. Electrical resistivity decrease sharply.

Factors influence recrystallization behavior are (1) Amount of deformation

(2) temperature (3) time (4) initial grain size (5) composition (6) amount of

recovery or polygonisation (7) Method of deformation. Hence

recrystallization temperature is not a fixed temperature in the sense of a

melting temperature. It can be defined as the temperature at which a given

alloy in a highly cold-worked state completely recrystallizes in 1h. The laws

of recrystallization are: (1) a minimum amount of deformation is needed to

cause recrystallization. (2) Smaller the degree of deformation, higher the

temperature required to cause recrystallization. (3) Recrystallization rate

increases exponentially with temperature. Doubling the annealing time is

approximately equivalent to increasing the annealing temperature 10°C. (4)Greater degree of deformation and lower annealing temperature, the smaller

the recrystallized grains. (5) Larger the original grain size, the greater the

amount of cold-work required to produce equivalent recrystallization

temperature. (6) The recrystallization temperature decreases with increasing

impurity of motel. Alloying always raise recrystallization temperature. (7)

The amount of deformation required to produce equivalent recrystallization

behavior increases with increased temperature of working.

Solute and Pinning effects The impurity in metal segregate at grain

boundary and retard the migrating boundaries during recrystallization. This

is known as the solution drag effect. When fine second phase particle

(carbides) lies on the migrating boundary, the grain boundary area is

reduced by an amount equal to cross sectional area of particle. When the

boundary moves further, it has to pull away from the particle and thereby

create new boundary are equal to cross sectional area of particle. This

increases energy and manifests itself as a pinning acting on the boundary.

Consequently the rate of recrystallization decreases.

Grain growth It is uniform increase in the average grain size following

recrystallization. The grain size distribution does not change during normal

grain growth. During abnormal grain growth called secondary

recrystallization because the phenomenon shows kinetics similar to

recrystallization, the grain size distribution may radically change i.e. some

very large grains present along with the fine grains. The driving force for

abnormal growth is decrease in surface energy. Solute drag and pinningaction of second phase particles retard movement of a migrating boundary

during grain growth as well.

comparison of mechanical properties during Recovery, Recrystallization and Grain

growth.


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